#326720

order	512
nilpotency class	3
isoclinism	isoclinic class of #326683
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23930	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,16)(11,12,15,13)(17,18,21,25)(19,27,30,23)(20,24,22,26)(28,31,29,32);(2,5)(6,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,29)(21,30)(22,28)(24,31)(25,27)(26,32);(1,3)(2,6)(4,7)(5,8)(10,15)(11,16)(12,13)(18,24)(19,28)(20,22)(25,26)(29,30)(31,32);(3,7)(6,8)(17,20,21,22)(18,24,25,26)(19,29,30,28)(23,31,27,32)

Polycyclic presentation:

2^9|a^2=h,d^2=hi,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=de,f^c=fh,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233916);